On the Integrability of (2+1)-Dimensional Quasilinear Systems

Abstract

A (2+1)-dimensional quasilinear system is said to be 'integrable' if it can be decoupled in infinitely many ways into a pair of compatible n-component onedimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogs of n-gap solutions. It is demonstrated that the requirement of the existence of 'sufficiently many' n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.MSC: 35L40, 35L65, 37K10.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

“…This property was used in [12,13](see also [48], [14], [20], [34]) when introducing the integrability criterion for a wide class of kinetic equations, corresponding hydrodynamic chains and 2+1 quasilinear equations. Moreover, it was proved in [36] that the existence of at least one N-component hydrodynamic reduction written in the so-called symmetric form is sufficient for integrability in the sense of [12]. Another possible approach to analyse an integrable kinetic equation is to use the fact that it possesses infinitely many particular solutions determined by the corresponding hydrodynamic reductions (see [34] for details).…”

“…Here the matrix ∆ (12) ik is the matrix ∆ with first two rows and ith and kth columns deleted. In the derivation of (93) …”

“…While the studied 'cold-gas' reductions turn out to be integrable for an arbitrary symmetric 'interaction kernel' G(η, µ), integrability of the full equation (1) will clearly require some additional restrictions to be imposed on this kernel. Recent results [12,13], [18,19] on the integrability of 2+1 hydrodynamic type systems and hydrodynamic chains, which are close 'relatives' of kinetic equations, suggest that these restrictions should be determined by the condition of the existence, for an arbitrary N, of N-component hydrodynamic reductions parameterised by N arbitrary functions of a single variable. The most natural way to attack this problem is to study the associated hydrodynamic chain, i.e.…”

The classification of nonsingular multidimensional Dubrovin-Novikov brackets

“…However, the case of multidimensional systems of hydrodynamic type is considerably more complicated and, in fact, has not been studied as yet from the viewpoint of the general Hamiltonian and integrability properties (see [5]). In the multidimensional case, very serious problems arise already under studying the simplest natural invariant class of suitable local Hamiltonian structures (1.1).…”

Characteristic Lie rings, finitely-generated modules and integrability conditions for (2 + 1)-dimensional lattices

“…A set of -integrals (1) , (2) , ... ( ) constitutes a complete set of integrals if none of these integrals is expressed in terms of the other ones and their total derivatives with respect to . In a similar way the complete set of -integrals for (2.1) is defined.…”